On Timoshenko-Reissner type theories of plates and shells

Gol'denveizer, A.L.; Kaplunov, Julius; Nolde, E.V.

doi:10.1016/0020-7683(93)90029-7

Cited by 103 publications

(127 citation statements)

References 12 publications

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“…We apply to the boundary value problem for the coating (2.1), (2.2), (2.4) and (3.1) the standard scheme of direct asymptotic integration (e.g. Goldenveizer et al 1993 and references therein). Let us specify a small parameter…”

Section: Effective Boundary Conditionsmentioning

confidence: 99%

A long-wave model for the surface elastic wave in a coated half-space

2010

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The paper deals with the three-dimensional problem in linear isotropic elasticity for a coated half-space. The coating is modelled via the effective boundary conditions on the surface of the substrate initially established on the basis of an ad hoc approach and justified in the paper at a long-wave limit. An explicit model is derived for the surface wave using the perturbation technique, along with the theory of harmonic functions and Radon transform. The model consists of three-dimensional 'quasi-static' elliptic equations over the interior subject to the boundary conditions on the surface which involve relations expressing wave potentials through each other as well as a two-dimensional hyperbolic equation singularly perturbed by a pseudo-differential (or integro-differential) operator. The latter equation governs dispersive surface wave propagation, whereas the elliptic equations describe spatial decay of displacements and stresses. As an illustration, the dynamic response is calculated for impulse and moving surface loads. The explicit analytical solutions obtained for these cases may be used for the non-destructive testing of the thickness of the coating and the elastic moduli of the substrate.

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Section: Effective Boundary Conditionsmentioning

confidence: 99%

A long-wave model for the surface elastic wave in a coated half-space

2010

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“…Therefore, one may expect that the first cut-offfrequency being small leads to the phenomena of weak coupling between the fundamental mode and the first harmonic, requiring special shear deformation plate theories in the long-wave low-frequency region, see e.g. [12, 13,14].…”

Section: Global Low-frequency Regimementioning

confidence: 99%

Multi-parametric analysis of the lowest natural frequencies of strongly inhomogeneous elastic rods

Kaplunov

Приказчиков

Sergushova

2016

Journal of Sound and Vibration

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The results of a multi-parametric analysis of the near-rigid body motions of a three-component strongly inhomogeneous elastic rod are presented. It is demonstrated that the values of the associated lowest natural frequencies tend to zero at large/small ratios of material and geometric parameters. The low-frequency behavior is classified into global and local regimes and the general conditions supporting global low-frequency regimes are derived. As an example, a rod with piecewise uniform properties is considered. A perturbation procedure oriented to a more general setup is developed.

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“…For the sake of definiteness, we assume that the non-local behaviour is modelled by an exponential kernel involving a small internal lengthscale. In §3, we proceed with a long-wave asymptotic scheme, originating from Goldenveizer et al [30] and later developed by, for example, Dai et al [31] and Aghalovyan [32]. Within the framework of these studies, the characteristic wavelength is assumed to be much greater than a typical microscale parameter.…”

Section: Introductionmentioning

confidence: 99%

Refined boundary conditions on the free surface of an elastic half-space taking into account non-local effects

Chebakov¹,

Kaplunov²,

Rogerson³

2016

Proc. R. Soc. A.

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The dynamic response of a homogeneous halfspace, with a traction-free surface, is considered within the framework of non-local elasticity. The focus is on the dominant effect of the boundary layer on overall behaviour. A typical wavelength is assumed to considerably exceed the associated internal lengthscale. The leading-order long-wave approximation is shown to coincide formally with the 'local' problem for a half-space with a vertical inhomogeneity localized near the surface. Subsequent asymptotic analysis of the inhomogeneity results in an explicit correction to the classical boundary conditions on the surface. The order of the correction is greater than the order of the better-known correction to the governing differential equations. The refined boundary conditions enable us to evaluate the interior solution outside a narrow boundary layer localized near the surface. As an illustration, the effect of nonlocal elastic phenomena on the Rayleigh wave speed is investigated.

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On Timoshenko-Reissner type theories of plates and shells

Cited by 103 publications

References 12 publications

A long-wave model for the surface elastic wave in a coated half-space

A long-wave model for the surface elastic wave in a coated half-space

Multi-parametric analysis of the lowest natural frequencies of strongly inhomogeneous elastic rods

Refined boundary conditions on the free surface of an elastic half-space taking into account non-local effects

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